CBMS Regional Conference in the Mathematical Sciences - "Finite Morse Index Solutions and Related Topics" -Winter 2007

讲座将涉及有界域上非线性椭圆方程的边值问题。它们将涵盖当前感兴趣的几个重要主题，如分岔理论、域形状、莫尔斯指数解和峰值解；它出现在数学的各种应用中，如数学生物学、催化理论、材料科学和燃烧。还将讨论这些主题之间的相互关系。讲座将展示峰解理论和有限摩尔斯指数解理论之间的紧密联系。有限莫尔斯指数解对于研究非线性快速增长（大于临界增长）的非线性椭圆方程是非常有用的工具，这一点越来越明显。会谈还将包括一些有趣的开放问题，这些问题可能会引起该领域研究人员的兴趣。

The lectures will involve boundary-value problems for nonlinear elliptic equations on bounded domains. They will encompass several important topics of current interest such as bifurcation theory, domain shape, Morse index solutions, and peak solutions; which occur in a great variety of applications of mathematics such as mathematical biology, catalysis theory, material science, and combustion. There will also be a discussion of interrelationships between these topics.The lectures will show a close connection between the theory of peak solutions and the theory of finite Morse index solutions. It is becoming increasingly apparent that finite Morse index solutions are very useful tools for studying nonlinear elliptic equations with rapid growth in the nonlinearity (faster than the critical growth). The talks will also include a number of interesting open problems that may be of interest to researchers in the area.